Three solutions for pertubed Dirichilet problem

Abstract

In this paper we prove the existence of at least three distinct solutions to the following perturbed Dirichlet problem \begin{displaymath} \left\{ \begin{array}{ll} -\Delta u= f(x,u)+\lambda g(x,u) & \mbox{in\ } \Omega\\ u=0 & \mbox{on\ } \partial \Omega, \end{array}\right. \end{displaymath} where $\Omega\subset\mathbb{R}^N$ is an open bounded set with smooth boundary $\partial \Omega$ and $\lambda\in \mathbb{R}$. Under very mild conditions on $g$ and some assumptions on the behaviour of the potential of $f$ at $0$ and $+\infty$, our result assures the existence of at least three distinct solutions to the above problem for $\lambda$ small enough. Moreover such solutions belong to a ball of the space $W_0^{1,2}(\Omega)$ centered in the origin and with radius not dependent on $\lambda$. \end{abstract}